Evaluating volumes for magma chambers and magma withdrawn for caldera collapse

We develop an analytical model to infer the total volume of a magma chamber associated with caldera collapse and the critical volume of magma that must be withdrawn to induce caldera collapse. The diameter of caldera border fault, depth to the magma chamber, and volumes of magma erupted before the onset of collapse and of entire eruption are compiled for 14 representative calderas. The volume of erupted magma at the onset of collapse aligns between the total erupted volume of the other representative caldera-forming eruptions and the volume of eruptions without collapse during the post-caldera stage, correlating with the structural diameter of the calderas. The total volume of magma chamber is evaluated using a piston-cylinder collapse model, in which the competition between the decompression inside magma chamber and friction along the caldera fault controls the collapse. Estimated volumes of the magma chambers associated with caldera collapse are 3-10 km(3) for Vesuvius 79 A.D. to 3000-10 500 km(3) for Long Valley, correlating with the cube of caldera diameters. The estimated volumes of magma chamber are always larger than the total volume of erupted magma for caldera formation, suggesting that the magma chambers are never completely emptied by the caldera-forming eruptions. The minimum volumes of erupted magma to trigger collapse are calculated from the correlation between the caldera diameters and the evaluated volume of magma chambers. The minimum eruptive volume for the collapse correlates with the square of the caldera radius r and the square of the depth to the magma chamber h, and inversely correlates with the bulk modulus of magma, which is mainly controlled by the bubble fraction in the magma. A bubble fraction between 5 and 10% at the onset of collapse may explain the distribution of the erupted volumes at the onset of collapse of the calderas in nature. (C) 2014 Elsevier B.V. All rights reserved.